Theorem rpnnen1lem5 13033

Description: Lemma for rpnnen1 13035. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rpnnen1lem.1	⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1lem.2	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
rpnnen1lem.n	⊢ ℕ ∈ V
rpnnen1lem.q	⊢ ℚ ∈ V

Assertion

Ref	Expression
rpnnen1lem5	⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)

Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛

Allowed substitution hints:   𝑇(𝑥, 𝑘)

Proof of Theorem rpnnen1lem5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpnnen1lem.1	. . . 4 ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
2		rpnnen1lem.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
3		rpnnen1lem.n	. . . 4 ⊢ ℕ ∈ V
4		rpnnen1lem.q	. . . 4 ⊢ ℚ ∈ V
5	1, 2, 3, 4	rpnnen1lem3 13031	. . 3 ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)
6	1, 2, 3, 4	rpnnen1lem1 13030	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ))
7	4, 3	elmap 8881	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (ℚ ↑m ℕ) ↔ (𝐹‘𝑥):ℕ⟶ℚ)
8	6, 7	sylib 221	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥):ℕ⟶ℚ)
9		frn 6714	. . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℚ)
10		qssre 13011	. . . . . 6 ⊢ ℚ ⊆ ℝ
11	9, 10	sstrdi 3946	. . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℝ)
12	8, 11	syl 18	. . . 4 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ⊆ ℝ)
13		1nn 12271	. . . . . . . 8 ⊢ 1 ∈ ℕ
14	13	ne0ii 4293	. . . . . . 7 ⊢ ℕ ≠ ∅
15		fdm 6716	. . . . . . . 8 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) = ℕ)
16	15	neeq1d 3016	. . . . . . 7 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (dom (𝐹‘𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
17	14, 16	mpbiri 261	. . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) ≠ ∅)
18		dm0rn0 5912	. . . . . . 7 ⊢ (dom (𝐹‘𝑥) = ∅ ↔ ran (𝐹‘𝑥) = ∅)
19	18	necon3bii 3009	. . . . . 6 ⊢ (dom (𝐹‘𝑥) ≠ ∅ ↔ ran (𝐹‘𝑥) ≠ ∅)
20	17, 19	sylib 221	. . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ≠ ∅)
21	8, 20	syl 18	. . . 4 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ≠ ∅)
22		breq2 5111	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ 𝑥))
23	22	ralbidv 3187	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
24	23	rspcev 3579	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦)
25	5, 24	mpdan 700	. . . 4 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦)
26		id 23	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27		suprleub 12208	. . . 4 ⊢ (((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
28	12, 21, 25, 26, 27	syl31anc 1400	. . 3 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
29	5, 28	mpbird 260	. 2 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥)
30	1, 2, 3, 4	rpnnen1lem4 13032	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
31		resubcl 11549	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
32	30, 31	mpdan 700	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
33	32	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
34		posdif 11734	. . . . . . . . . 10 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
35	30, 34	mpancom 701	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
36	35	biimpa 482	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )))
37	36	gt0ne0d 11805	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ≠ 0)
38	33, 37	rereccld 12069	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∈ ℝ)
39		arch 12528	. . . . . 6 ⊢ ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘)
40	38, 39	syl 18	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘)
41	40	ex 418	. . . 4 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘))
42	1, 2	rpnnen1lem2 13029	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
43	42	zred 12728	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44	43	3adant3 1150	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
45	44	ltp1d 12172	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
46	33, 36	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
47		nnre 12267	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48		nngt0 12294	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
49	47, 48	jca 521	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50		ltrec1 12129	. . . . . . . . . . . . 13 ⊢ ((((𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
51	46, 49, 50	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
52	30	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
53		nnrecre 12305	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
54	53	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
56	52, 54, 55	ltaddsub2d 11842	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
57	12	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹‘𝑥) ⊆ ℝ)
58		ffn 6706	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (𝐹‘𝑥) Fn ℕ)
59	8, 58	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) Fn ℕ)
60		fnfvelrn 7076	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥))
61	59, 60	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥))
62	57, 61	sseldd 3935	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ℝ)
63	30	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
64	53	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
65	12, 21, 25	3jca 1146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦))
66	65	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦))
67		suprub 12203	. . . . . . . . . . . . . . . . 17 ⊢ (((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) ∧ ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥)) → ((𝐹‘𝑥)‘𝑘) ≤ sup(ran (𝐹‘𝑥), ℝ, < ))
68	66, 61, 67	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ≤ sup(ran (𝐹‘𝑥), ℝ, < ))
69	62, 63, 64, 68	leadd1dd 11855	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)))
70	62, 64	readdcld 11265	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71		readdcl 11210	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
72	30, 53, 71	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74		lelttr 11327	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
75	74	expd 421	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
76	70, 72, 73, 75	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
77	69, 76	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
78	77	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
79	56, 78	sylbird 263	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
80	51, 79	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
81	42	peano2zd 12731	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82		oveq1 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
83	82	breq1d 5117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
84	83, 1	elrab2 3652	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(𝑇, ℝ, < ) + 1) ∈ 𝑇 ↔ ((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
85	84	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ (((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
86	81, 85	sylan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87		ssrab2 4031	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
88	1, 87	eqsstri 3980	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 ⊆ ℤ
89		zssre 12625	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ ⊆ ℝ
90	88, 89	sstri 3943	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ⊆ ℝ
91	90	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92		remulcl 11212	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
93	92	ancoms 464	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
94	47, 93	sylan2 605	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95		btwnz 12727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
96	95	simpld 500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
97	94, 96	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98		zre 12622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
99	98	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100		simpll 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
101	49	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102		ltdivmul 12117	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥 ↔ 𝑛 < (𝑘 · 𝑥)))
103	99, 100, 101, 102	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥 ↔ 𝑛 < (𝑘 · 𝑥)))
104	103	rexbidva 3186	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥 ↔ ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥)))
105	97, 104	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
106		rabn0 4342	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅ ↔ ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
107	105, 106	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
108	1	neeq1i 3021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109	107, 108	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
110	1	reqabi 3437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
111	47	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112	111, 100, 92	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113		ltle 11325	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
114	99, 112, 113	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
115	103, 114	sylbid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥 → 𝑛 ≤ (𝑘 · 𝑥)))
116	115	impr 460	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117	110, 116	sylan2b 606	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118	117	ralrimiva 3156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥))
119		breq2 5111	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑘 · 𝑥) → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ (𝑘 · 𝑥)))
120	119	ralbidv 3187	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑘 · 𝑥) → (∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121	120	rspcev 3579	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦)
122	94, 118, 121	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦)
123	91, 109, 122	3jca 1146	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦))
124		suprub 12203	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125	123, 124	sylan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
126	86, 125	syldan 603	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
127	126	ex 418	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < )))
128	42	zcnd 12729	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129		1cnd 11229	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130		nncn 12268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131		nnne0 12297	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132	130, 131	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133	132	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134		divdir 11924	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝑇, ℝ, < ) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
135	128, 129, 133, 134	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
136	3	mptex 7225	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
137	2	fvmpt2 7002	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹‘𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138	136, 137	mpan2 704	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139	138	fveq1d 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → ((𝐹‘𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140		ovex 7449	. . . . . . . . . . . . . . . . . 18 ⊢ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141		eqid 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142	141	fvmpt2 7002	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143	140, 142	mpan2 704	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144	139, 143	sylan9eq 2817	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145	144	oveq1d 7431	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146	135, 145	eqtr4d 2800	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)))
147	146	breq1d 5117	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
148	81	zred 12728	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149	148, 43	lenltd 11383	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ) ↔ ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
150	127, 147, 149	3imtr3d 296	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
151	150	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
152	80, 151	syld 48	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
153	152	exp31 425	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
154	153	com4l 93	. . . . . . . 8 ⊢ (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (𝑥 ∈ ℝ → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
155	154	com14 97	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
156	155	3imp 1128	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
157	45, 156	mt2d 137	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)
158	157	rexlimdv3a 3169	. . . 4 ⊢ (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥))
159	41, 158	syld 48	. . 3 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥))
160	159	pm2.01d 192	. 2 ⊢ (𝑥 ∈ ℝ → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)
161		eqlelt 11324	. . 3 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)))
162	30, 161	mpancom 701	. 2 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)))
163	29, 160, 162	mpbir2and 726	1 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107   ↦ cmpt 5190  dom cdm 5659  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  supcsup 9413  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130   · cmul 11132   < clt 11270   ≤ cle 11271   − cmin 11468   / cdiv 11898  ℕcn 12260  ℤcz 12618  ℚcq 13000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-n0 12532  df-z 12619  df-q 13001

This theorem is used by:  rpnnen1lem6  13034